Critical Exponents for Long-Range $${O(n)}$$ O ( n ) Models Below the Upper Critical Dimension

Abstract

We consider the critical behaviour of long-range O(n) models (n ≥ 0) on Z d , with interaction that decays with distance r as r −(d+α) , for α ∈ (0, 2). For n ≥ 1, we study the n-component |ϕ| 4 lattice spin model. For n = 0, we study the weakly self-avoiding walk via an exact representation as a supersymmetric spin model. These models have upper critical dimension d c = 2α. For dimensions d = 1, 2, 3 and small ǫ > 0, we choose α = 1 2 (d + ǫ), so that d = d c − ǫ is below the upper critical dimension. For small ǫ and weak coupling, to order ǫ we prove existence of and compute the values of the critical exponent γ for the susceptibility (for n ≥ 0) and the critical exponent α H for the specific heat (for n ≥ 1). For the susceptibility, γ = 1 + n+2 n+8 ǫ α + O(ǫ 2 ), and a similar result is proved for the specific heat. Expansion in ǫ for such long-range models was first carried out in the physics literature in 1972. Our proof adapts and applies a rigorous renormalisation group method developed in previous papers with Bauerschmidt and Brydges for the nearest-neighbour models in the critical dimension d = 4, and is based on the construction of a non-Gaussian renormalisation group fixed point. Some aspects of the method simplify below the upper critical dimension, while some require different treatment, and new ideas and techniques with potential future application are introduced.

Critical Two-Point Function for Long-Range O(n) Models Below the Upper Critical Dimension

“…This proves the "sticking" of the critical exponent at its mean-field value, for α slightly above d 2 , or equivalently, for d slightly below the upper critical dimension d c = 2α. Our proof extends recent results and methods used to study the ǫ-expansion for the critical exponents for the susceptibility and specific heat of the long-range models [31]. It also relies on results and techniques developed to study related problems for the 4-dimensional nearest-neighbour models [5,16,32].…”

“…We now define the fractional Laplacian and list some of its properties. Further details can be found in [31,.…”

“…For |x − y| → ∞, the fractional Laplacian decays as − (−∆) α/2 x,y ≍ |x − y| −(d+α) (1.4) (see [31,Lemma 2.1], or [10,Theorem 5.3] for a more precise and more general statement). Here, and in the following, we write a ≍ b to denote the existence of c > 0 such that c −1 b ≤ a ≤ cb.…”

“…Let X denote the continuous-time Markov chain with state space Z d and infinitesimal generator Q = −(−∆ Z d ) α/2 . Verification that Q has the attributes required of a generator is given in [31,Lemma 2.4]. Let P be the probability measure associated with X, and E the corresponding expectation; a subscript a specifies X(0) = a.…”

“…The labels 0 on the left-hand sides of (1.16)-(1.17) reflect the fact that the weakly self-avoiding walk corresponds to the formal n = 0 case of the n-component |ϕ| 4 model. As in earlier work on the 4-dimensional case, e.g., [31,32], we treat both cases n ≥ 1 (spins) and n = 0 (self-avoiding walk) simultaneously and rigorously, via a supersymmetric spin representation for the weakly self-avoiding walk.…”